SOLUTION: A box with a square top and bottom is to be made to contain a volume of 64 cubic inches. What should be the dimensions of the box if its surface area is to be minimum? What is this

Question 1155310: A box with a square top and bottom is to be made to contain a volume of 64 cubic inches. What should be the dimensions of the box if its surface area is to be minimum? What is this minimum surface area?
Answer by Cromlix(4381) (Show Source): You can put this solution on YOUR website!
Hi,
Taking the measurements of the length and breadth of the base and top of the box = x
Taking it to be a closed box (Closed top and bottom)
Volume = x^2 h (h being height)
Volume = 64 ins^3
x^2 h = 64
Therefore h = 64/x^2
Surface area = 2x^2 (top and bottom)+ 4xh (4 sides)
Surface Area = 2x^2 + 4xh
(Removing the h) by multiplying 4xh by 64/x^2 (h)
4xh * 64/x^2 ( * means times)
= 256/x
Surface Area = 2x^2 + 256/x
S.A.(x) = 2x^2 + 256x^-1
S.A.'(x)= 4x - 256x^-2
S.A.' (x) = 4x - 256/x^2
S.A.' (x) = 0
4x - 256/x^2 = 0
- 256/x^2 = - 4x (Multiply both sides by -1)
256/x^2 = 4x (Cross multiply)
256 = 4x^3
4x^3 = 256
x^3 = 64
x = cube root of 64
x = 4
Nature Table shows x = 4 to be a minimum value
Therefore Length = Breadth = 4 ins
Height = 64/x^2 = 64/16 = 4 ins.
Hope this helps :-)

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